SAFPWR home page

Safpwr neutron kinetics : definition of planar reactivity.

In analogy with the definition of static core reactivity, it is possible to define a planar dynamic reactivity ρ from (1a) where it is reminded that r includes the radial leakage but excludes axial leakage; φ is the searched dynamic flux and ψ remains, up now, undefined.
(1a) may be written as (1b).
Substituting in the general form (37), and noting (1c) leads to (4a).
From ((37b), we get (4b).
In practice, if the axial leakage can be neglected in the thermal balance, the model can be reduced to (4c).
This attractively simple form has no practical interest unless ρ can be identified with the static planar reactivity as could be obtained from the reactor nuclear design code fed by the current values of assembly-wise macroscopic cross-sections.
For this to be true, the necessary and sufficient condition is that ψ identifies with the solution of the adjoint form of the same static problem.
Indeed, if ηz(x,y,V) represents the solution of the planar static eigen-value problem, it must verify the eigen-value equation (7a) at each point of plane z;
its adjoint form (7b) possesses the same eigen-values spectrum.
Adjoint operators are defined by means of the commutation relation (7c) which must hold for any fields φ and ψ, with the sole restriction that they satisfy the same internal (inter nodal) and boundary conditions as those imposed to the direct flux.
Thus, particularly, (10a) must hold for any field φ and ψ or (10b), from (7a ) and (7b) which show well that under those conditions, static and dynamic reactivity identify (10d).

The obvious interest of adjoint flux weighing is to select, as reduction invariant, the most significant parameter of core physical state, namely the static reactivity in each of its planes.

In practice, one-group model parameters must be obtained from the known static flux η rather than from the unknown dynamic flux φ (12, 16) where neutron densities are calculated from the group fluxes (17).
Introducing logarithmic time derivative (18), the one-group equations may finally write as (19), which clearly shows that the core dynamic behavior is governed by the fission operator F and by the group (19e).
Actually the correction Λ ω to β~ in (19e) acts as a "fuse" preventing (19e) of becoming singular as planar reactivity approaches and exceeds β~

Group 2 correction.

General fast neutron balance can be recast as (33a), where the second line represents the thermal neutrons correction term.
In the thermal neutron balance (33b) for each (x,y,v) element, (σrφ)1 represents the inscattering rate from fast neutrons and (σφ)2 is the net removal rate by scattering, absorption and radial leakage.
Defining, as usual, planar averages from (33d:g) and (34a:d), and corrected values from (33e:h) allows representing (35a) the thermal correction to the fast balance as an additional term depending of the thermal to fast flux ratio.
This flux ratio can be viewed as a small additional feed-back correction calculated from the results of the thermal balance (35b)
If, in (35b), both N2 time derivative and leakage terms are neglected, then we come back to the 1-group approximation.
Generally, the time derivative term is much smaller than the leakage one and the flux ratio can then be calculated from a single diffusion axial sweeping made at eos, and taken as constant in the course of the fast neutron iterations of the subsequent step.

Asymptotic solutions

One group model

We look for an exponential solution to the effective one-group system (21) where exponent "1" to the fast values has been dropped.
Substituting (22a,b) in (21a) and expressing planar reactivity from (22e) we obtain, after little algebra, the relation (22d) which mean that, if Λ and the β~ are the same in all core planes, then the relation (22d) relates the eigen-value to the core logarithmic derivative.
Or, in terms of static core reactivity ρc, we come back to the same Nordheim relation (22f) as in point kinetics, because static flux verifies eigen-value equation (29)

2-group asymptotic relation

Substituting in (61-63) an exponential solution of log derivative ω for both Φ and Cj leads to the homogeneous system (30a,b) with the core eigen-value given by (30c), just like in the 1-group model, except that λc* is no longer the core static eigen-value λc, solution of (30d,e).
It is feasible to obtain a solution of (30a,b) by iterations.

This solution is precious for benchmarking 2-group, 1-D, transient resolution for heterogeneous cases.